A Note On Impossibility Theorems and Seniority Rules

We characterize seniority rules, also known as lexical dictatorships, under weak consistency constraints on the groups choice function. These constraints are base triple-acyclicity in the case of binary choices and rationalizability (although not rationality) in the case of choices between an arbitrary number of alternatives. Existing results on these weakened constraints remain silent on the treatment of the groups most junior individuals and therefore do not yield a complete characterization of seniority rules. We also impose a universal domain, binary strict Pareto optimality, binary Pareto indifference, binary independence of irrelevant alternatives, and the newly introduced condition of conflict resolution. The latter condition requires a social choice rules not to remain indecisive between alternatives for which individuals have conflicting preferences.JEL CLASSIFICATIONS: D63, D71This revised version was published online in May 2005 with a corrected article title.     

Unanimity and Resource Monotonicity

In the context of indivisible public objects problems (e.g., candidate selection or qualification) with “separable” preferences, unanimity rule accepts each object if and only if the object is in everyone’s top set. We establish two axiomatizations of unanimity rule. The main axiom is resource monotonicity, saying that resource increase should affect all agents in the same direction. This axiom is considered in combination with simple Pareto (there is no Pareto improvement by addition or subtraction of a single object), independence of irrelevant alternatives, and either path independence or strategy-proofness.     

Randomized dictatorship and the Kalai–Smorodinsky bargaining solution

“Randomized dictatorship,” one of the simplest ways to solve bargaining situations, works as follows: a fair coin toss determines the “dictator”—the player to be given his first-best payoff. The two major bargaining solutions, that of Nash (Econometrica 18:155–162, 1950) and that of Kalai and Smorodinsky (Econometrica, 43:513–518, 1975), Pareto-dominate this process (in the ex ante sense). However, whereas the existing literature offers axiomatizations of the Nash solution in which this ex ante domination plays a central role (Moulin, Le choix social utilitariste, Ecole Polytechnique Discussion Paper, 1983 ; de Clippel, Social Choice and Welfare, 29:201–210, 2007), it does not provide an analogous result for Kalai–Smorodinsky. This paper fills in this gap: a characterization of the latter is obtained by combining the aforementioned domination with three additional axioms: Pareto optimality, individual monotonicity, and a weakened version of the Perles–Maschler (International Journal of Game Theory, 10:163–193, 1981) super additivity axiom.     

Risk aversion in n-person bargaining

Relations between risk properties of Pareto optimal n-person bargaining solutions are studied. The (weak) worse alternative property, risk sensitivity property, and risk profit opportunity property are considered, in particular relations between these properties, and between these properties and other properties such as independence of irrelevant alternatives and individual monotonicity. A distinction is made between bargaining games where all Pareto optimal outcomes are riskless, and bargaining games where Pareto optimal outcomes may represent the utilities of lotteries between riskless alternatives. In the first mentioned case, more general results can be obtained.     

Independence of irrelevant alternatives revisited

This paper aims to reexamine the axiom of the independence of irrelevant alternatives in the theory of social choice. A generalized notion of independence is introduced to clarify an informational requirement of binary independence which is usually imposed in the Arrovian framework. We characterize the implication of binary independence.     

A theorem for Bayesian group decisions

This paper presents a natural extension of Bayesian decision theory from the domain of individual decisions to the domain of group decisions. We assume that each group member accepts the assumptions of subjective expected utility theory with respect to the alternatives from which they must choose, but we do not assume, a priori, that the group as a whole accepts those assumptions. Instead, we impose a multiattribute utility independence condition on the preferences of the group with respect to the expected utilities of its actions as appraised by its members. The result is that the expected utility of an alternative for the group is a weighted average of the expected utilities of that alternative for its members. The weights must be determined collectively by the group. Pareto optimality is not assumed, though the result is consistent with Pareto optimality.     

Divide the Dollar: Three solutions and extensions

Divide the Dollar (DD) is a game in which two players independently bid up to 100 cents for a dollar. Each player receives his or her bid if the sum of the bids does not exceed a dollar; otherwise they receive nothing. This game has multiple Nash equilibria, including the egalitarian division of (50, 50), but this division is not compelling except for its symmetry and presumed fairness.This division is easy to induce, however, by punishing — more severely than does DD — deviations from it, but these solutions are not reasonable. By altering the rules of DD, however, one can induce an egalitarian division (by successive elimination of weakly dominated strategies), but no reasonable payoff scheme produces this division with egalitarian bids of 50.Three alternatives to DD are analyzed. DD1, which rewards lowest bidders first, shows how an egalitarian outcome can be induced with equal but nonegalitarian bids. DD2, which adds a second stage that provides the players with new information yet restricts their choices at the same time, is used to introduce dominance inducibility. DD3 combines the features of DD1 and DD2, is reasonable (like DD1), makes calculations transparent (like DD2), and induces egalitarian bids as well as the egalitarian outcome. The possible application of the different procedures to a real-world allocation problem (setting of salaries by a team), in which there may be entitlements, is described.     

An axiomatic analysis of the Nash equilibrium concept

The purpose of this paper is to analyze axiomatically the Nash equilibrium concept. The class of games under study is a (relatively large) subclass of n-person normal form games. Solutions are correspondences which associate to each game a non empty set of strategy vectors of this game. It is shown that if a solution satisfies the axioms Independence of irrelevant alternatives (IIA) and Individual rationality (IR), then all the strategy vectors in this solution are Nash equilibria. This result holds good also if IR is replaced by Strong individual monotonicity (SIM) or Weak principle of fair compromise (WPFC).     

How vicious are cycles of intransitive choice?

Transitivity is a compelling requirement of rational choice, and a transitivity axiom is included in all classical theories of both individual and group choice. Nonetheless, choice contexts exist in which choice might well be systematically intransitive. Moreover, this can occur even when the context is transparent, and the decision maker is reflective. The present paper catalogues such choice contexts, dividing them roughly into the following classes:

Resolute majority rules

We investigate resolute voting rules that always rank two alternatives strictly and avoid social indecision. Resolute majority rules differ from the standard majority rule in that whenever both alternatives win the same number of votes, a tie-breaking function is used to determine the outcome. We provide axiomatic characterizations of resolute majority rules or resolute majority rules with a quorum. Resoluteness axiom is used in all these results. The other axioms are weaker than those considered in the characterization of the majority rule by May (1952 Econometrica, 20:680–684). In particular, instead of May’s positive responsiveness, we consider a much weaker monotonicity axiom.     

Ordinal independence in nonlinear utility theory

Individual behavior under uncertainty is characterized using a new axiom, ordinal independence, which is a weakened form of the von Neumann-Morgenstern independence axiom It states that if two distributions share a tail in common, then this tail can be modified without altering the individual's preference between these distributions. Preference is determined by the tail on which the distributions differ. This axiom implies an appealing and simple functional form for a numerical representation of preferences. It generalizes the form of anticipated utility, and it explains some well-known forms of behavior, such as the Friedman-Savage paradox, that anticipated utility cannot.     

An additive representation on the product of complete,continuous extensive structures

This article develops an axiom system to justify an additive representation for a preference relation \({\succsim}\) on the product \({\prod_{i=1}^{n}A_{i}}\) of extensive structures. The axiom system is basically similar to the n-component (n ≥ 3) additive conjoint structure, but the independence axiom is weakened in the system. That is, the axiom exclusively requires the independence of the order for each of single factors from fixed levels of the other factors. The introduction of a concatenation operation on each factor A _i makes it possible to yield a special type of restricted solvability, i.e., additive solvability and the usual cancellation on \({\prod_{i=1}^{n}A_{i}}\). In addition, the assumption of continuity and completeness for A _i implies a stronger type of solvability on A _i . The additive solvability, cancellation, and stronger solvability axioms allow the weakened independence to be effective enough in constructing the additive representation.     

Violations of betweenness and choice shifts in groups

In decision theory, the betweenness axiom postulates that a decision maker who chooses an alternative A over another alternative B must also choose any probability mixture of A and B over B itself and can never choose a probability mixture of A and B over A itself. The betweenness axiom is a weaker version of the independence axiom of expected utility theory. Numerous empirical studies documented systematic violations of the betweenness axiom in revealed individual choice under uncertainty. This paper shows that these systematic violations can be linked to another behavioral regularity—choice shifts in a group decision making. Choice shifts are observed if an individual faces the same decision problem but makes a different choice when deciding alone and in a group.     

Axiomatic characterizations of solutions for Bayesian games

Bayesian equilibria are characterized by means of consistency and one-person rationality in combination with non-emptiness or converse consistency. Moreover, strong and coalition-proof Bayesian equilibria of extended Bayesian games are introduced and it is seen that these notions can be characterized by means of consistency, one-person rationality, a version of Pareto optimality and a modification of converse consistency. It is shown that, in case of the strong Bayesian equilibrium correspondence, converse consistency can be replaced by non-emptiness. As examples we treat Bayesian potential games and Bayesian congestion games.     

Independent social choice correspondences

A fixed agenda social choice correspondence on outcome set X maps each profile of individual preferences into a nonempty subset of X. If satisfies an analogue of Arrow's independence of irrelevant alternatives condition, then either the range of contains exactly two alternatives, or else there is at most one individual whose preferences have any bearing on . This is the case even if is not defined for any proper subset of X.     

Security level,potential level,expected utility: A three-criteria decision model under risk

Among the violations of expected utility (E.U.) theory which have been observed by experimenters, the violations of its independence axiom is, by far, the most common. It seems that, in many cases, these inconsistencies can be ascribed to the desire for security - called the security factor by L. Lopes (1986) - which makes people attach special importance to the worst outcomes of risky decisions as well as to the sole outcomes of riskless decisions (certainty effect). J.-Y. Jaffray (1988) has proposed a model which generalizes E.U. theory by taking into account this factor and is then able to account for certain violations. However, especially in experiments on choice involving prospective losses, violations of the von Neumann-Morgenstern independence axiom cannot be explained by the security factor alone and have to be partially ascribed to the potential factor (L. Lopes, 1986) which reflects heightened attention to the best outcomes of decisions, especially when the best outcome is the status quo. In this paper, we construct an axiomatic model for subjects taking into account simultaneously or alternatively the security factor and the potential factor. For this, as in Jaffray's model, it has been necessary to weaken not only the standard independence axiom but also the continuity axiom and, in the same time, to reinforce the dominance axiom. In the resulting model, choices are partially determined by the mere comparison of the (security level, potential level) (i.e. the (worst outcome, best outcome)) pairs offered, and completed by the maximization of an affine function of the expected utility, the coefficients of which depend on both the security level and potential level.In this model, a decision maker who (i) has constant marginal utility for money, (ii) is sensitive to the security factor alone in the domain of gains, (iii) is sensitive to the potential factor alone in the domain of losses, behaves as a risk averter for gains and a risk seeker for losses.     

Implementing egalitarianism in a class of Nash demand games

We add a stage to Nash’s demand game by allowing the greedier player to revise his demand if the demands are not jointly feasible. If he decides to stick to his initial demand, then the game ends and no one receives anything. If he decides to revise it down to \(1-x\), where x is his initial demand, the revised demand is implemented with certainty. The implementation probability changes linearly between these two extreme cases. We derive a condition on the feasible set under which the two-stage game has a unique subgame perfect equilibrium. In this equilibrium, there is first-stage agreement on the egalitarian demands. We also study two n-player versions of the game. In either version, if the underlying bargaining problem is “divide-the-dollar,” then equal division is sustainable in a subgame perfect equilibrium if and only if the number of players is at most four.     

Axiomatization of a class of share functions for n-person games

The Shapley value is the unique value defined on the class of cooperative games in characteristic function form which satisfies certain intuitively reasonable axioms. Alternatively, the Banzhaf value is the unique value satisfying a different set of axioms. The main drawback of the latter value is that it does not satisfy the efficiency axiom, so that the sum of the values assigned to the players does not need to be equal to the worth of the grand coalition. By definition, the normalized Banzhaf value satisfies the efficiency axiom, but not the usual axiom of additivity.In this paper we generalize the axiom of additivity by introducing a positive real valued function on the class of cooperative games in characteristic function form. The so-called axiom of -additivity generalizes the classical axiom of additivity by putting the weight (v) on the value of the gamev . We show that any additive function determines a unique share function satisfying the axioms of efficient shares, null player property, symmetry and -additivity on the subclass of games on which is positive and which contains all positively scaled unanimity games. The axiom of efficient shares means that the sum of the values equals one. Hence the share function gives the shares of the players in the worth of the grand coalition. The corresponding value function is obtained by multiplying the shares with the worth of the grand coalition. By defining the function appropiately we get the share functions corresponding to the Shapley value and the Banzhaf value. So, for both values we have that the corresponding share functions belong to this class of share functions. Moreover, it shows that our approach provides an axiomatization of the normalized Banzhaf value. We also discuss some other choices of the function and the corresponding share functions. Furthermore we consider the axiomatization on the subclass of monotone simple games.     

Ambiguity framed

In his exposition of subjective expected utility theory, Savage (1954) proposed that the Allais paradox could be reduced if it were recast into a format which made the appeal of the independence axiom of expected utility theory more transparent. Recent studies consistently find support for this prediction. We consider a salience-based choice model which explains this frame-dependence of the Allais paradox. We then derive the novel prediction that the presentation format responsible for reductions in Allais-style violations of expected utility theory will also reduce Ellsberg-style violations of subjective expected utility theory. This format makes the appeal of Savage’s “sure thing principle” more transparent. We design an experiment to test this prediction and find strong support for such frame-dependence of ambiguity aversion in Ellsberg-style choices. In particular, we observe markedly less ambiguity-averse behavior in Savage’s matrix format than in a more standard “prospect” format. This finding poses a new challenge for the leading models of ambiguity aversion.